On the difference between the revised Szeged index and the Wiener index

Let Sz(G) and W (G) be the revised Szeged index and the Wiener index of a graph G. Chen, Li, and Liu [European J. Combin. 36 (2014) 237–246] proved that if G is a non-bipartite connected graph of order n ≥ 4, then Sz(G) −W (G) ≥ ( n + 4n− 6 ) /4. Using a matrix method we prove that if G is a non-bipartite graph of order n, size m, and girth g, then Sz(G)−W (G) ≥ n ( m− 3n 4 ) + P (g), where P is a fixed cubic polynomial. Graphs that attain the equality are also described. If in addition g ≥ 5, then Sz(G)−W (G) ≥ n ( m− 3n 4 ) + (n− g)(g − 3)+P (g). These results extend the bound of Chen, Li, and Liu as soon as m ≥ n+1 or g ≥ 5. The remaining cases are treated separately.